Users' Mathboxes Mathbox for Jeff Hoffman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  findabrcl Structured version   Unicode version

Theorem findabrcl 30899
Description: Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
findabrcl.1  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
Assertion
Ref Expression
findabrcl  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P
)
Distinct variable groups:    x, G    x, A    x, C    z, G    z, A    z, P
Allowed substitution hints:    C( z)    P( x)

Proof of Theorem findabrcl
StepHypRef Expression
1 elex 3096 . . . 4  |-  ( C  e.  om  ->  C  e.  _V )
2 fveq2 5881 . . . . 5  |-  ( x  =  C  ->  ( rec ( G ,  A
) `  x )  =  ( rec ( G ,  A ) `  C ) )
3 eqid 2429 . . . . 5  |-  ( x  e.  _V  |->  ( rec ( G ,  A
) `  x )
)  =  ( x  e.  _V  |->  ( rec ( G ,  A
) `  x )
)
4 fvex 5891 . . . . 5  |-  ( rec ( G ,  A
) `  C )  e.  _V
52, 3, 4fvmpt 5964 . . . 4  |-  ( C  e.  _V  ->  (
( x  e.  _V  |->  ( rec ( G ,  A ) `  x
) ) `  C
)  =  ( rec ( G ,  A
) `  C )
)
61, 5syl 17 . . 3  |-  ( C  e.  om  ->  (
( x  e.  _V  |->  ( rec ( G ,  A ) `  x
) ) `  C
)  =  ( rec ( G ,  A
) `  C )
)
76adantr 466 . 2  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  =  ( rec ( G ,  A ) `  C
) )
8 findabrcl.1 . . . 4  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
98findreccl 30898 . . 3  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
109imp 430 . 2  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( rec ( G ,  A ) `  C )  e.  P
)
117, 10eqeltrd 2517 1  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    |-> cmpt 4484   ` cfv 5601   omcom 6706   reccrdg 7135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator